
theorem Th3:
  multcomplex||REAL = multreal
  proof
    set mu = multcomplex||REAL;
    [:REAL,REAL:] c= [:COMPLEX,COMPLEX:] by NUMBERS:11,ZFMISC_1:96;
    then
A1: [:REAL,REAL:] c= dom(multcomplex) by FUNCT_2:def 1;
    then
A2: dom mu = [:REAL,REAL:] by RELAT_1:62;
A3: dom(multreal) = [:REAL,REAL:] by FUNCT_2:def 1;
    for z be object st z in dom mu holds mu.z = multreal.z
    proof
      let z be object;
      assume
A4:   z in dom mu;
      then consider x, y be object such that
A5:   x in REAL & y in REAL & z = [x,y] by A2,ZFMISC_1:def 2;
      reconsider x1 = x, y1 = y as Real by A5;
      thus mu.z = multcomplex.(x1,y1) by A4,A5,A2,FUNCT_1:49
      .= x1*y1 by BINOP_2:def 5
      .= multreal.(x1,y1) by BINOP_2:def 11
      .= multreal.z by A5;
    end;
    hence thesis by A1,A3,RELAT_1:62;
  end;
